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Mirrors > Home > MPE Home > Th. List > Mathboxes > 6gbe | Structured version Visualization version GIF version |
Description: 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
6gbe | ⊢ 6 ∈ GoldbachEven |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6even 43867 | . 2 ⊢ 6 ∈ Even | |
2 | 3prm 16030 | . . 3 ⊢ 3 ∈ ℙ | |
3 | 3odd 43864 | . . . 4 ⊢ 3 ∈ Odd | |
4 | gbpart6 43922 | . . . 4 ⊢ 6 = (3 + 3) | |
5 | 3, 3, 4 | 3pm3.2i 1334 | . . 3 ⊢ (3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = (3 + 3)) |
6 | eleq1 2898 | . . . . 5 ⊢ (𝑝 = 3 → (𝑝 ∈ Odd ↔ 3 ∈ Odd )) | |
7 | biidd 264 | . . . . 5 ⊢ (𝑝 = 3 → (𝑞 ∈ Odd ↔ 𝑞 ∈ Odd )) | |
8 | oveq1 7155 | . . . . . 6 ⊢ (𝑝 = 3 → (𝑝 + 𝑞) = (3 + 𝑞)) | |
9 | 8 | eqeq2d 2830 | . . . . 5 ⊢ (𝑝 = 3 → (6 = (𝑝 + 𝑞) ↔ 6 = (3 + 𝑞))) |
10 | 6, 7, 9 | 3anbi123d 1430 | . . . 4 ⊢ (𝑝 = 3 → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞)) ↔ (3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (3 + 𝑞)))) |
11 | biidd 264 | . . . . 5 ⊢ (𝑞 = 3 → (3 ∈ Odd ↔ 3 ∈ Odd )) | |
12 | eleq1 2898 | . . . . 5 ⊢ (𝑞 = 3 → (𝑞 ∈ Odd ↔ 3 ∈ Odd )) | |
13 | oveq2 7156 | . . . . . 6 ⊢ (𝑞 = 3 → (3 + 𝑞) = (3 + 3)) | |
14 | 13 | eqeq2d 2830 | . . . . 5 ⊢ (𝑞 = 3 → (6 = (3 + 𝑞) ↔ 6 = (3 + 3))) |
15 | 11, 12, 14 | 3anbi123d 1430 | . . . 4 ⊢ (𝑞 = 3 → ((3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (3 + 𝑞)) ↔ (3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = (3 + 3)))) |
16 | 10, 15 | rspc2ev 3633 | . . 3 ⊢ ((3 ∈ ℙ ∧ 3 ∈ ℙ ∧ (3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = (3 + 3))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞))) |
17 | 2, 2, 5, 16 | mp3an 1455 | . 2 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞)) |
18 | isgbe 43907 | . 2 ⊢ (6 ∈ GoldbachEven ↔ (6 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞)))) | |
19 | 1, 17, 18 | mpbir2an 709 | 1 ⊢ 6 ∈ GoldbachEven |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ∃wrex 3137 (class class class)co 7148 + caddc 10532 3c3 11685 6c6 11688 ℙcprime 16007 Even ceven 43780 Odd codd 43781 GoldbachEven cgbe 43901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-fz 12885 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-prm 16008 df-even 43782 df-odd 43783 df-gbe 43904 |
This theorem is referenced by: nnsum3primesle9 43950 |
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